From Barthel Randers Kropina Geometries to the Accelerating Universe: A Brief Review of Recent Advances in Finslerian Cosmology

TL;DR

This paper reviews recent advances in Finslerian cosmology, exploring how Finsler geometry extensions of general relativity can model the universe's accelerated expansion and fit observational data as an alternative to the standard cosmological model.

Contribution

It introduces new Finslerian cosmological models based on Barthel-Randers and Barthel-Kropina geometries, deriving generalized Friedmann equations and comparing their predictions with observations.

Findings

Finslerian models can effectively describe dark energy effects.

Models fit observational data as well as ΛCDM.

Additional geometric terms act as an effective cosmological constant.

Abstract

We review recent developments in cosmological models based on Finsler geometry and extensions of general relativity within this framework. Finsler geometry generalizes Riemannian geometry by allowing the metric tensor to depend on position and an additional internal degree of freedom, typically represented by a vector field at each point of the spacetime manifold. We explore whether Finsler-type geometries can describe gravitational interaction and cosmological dynamics. In particular, we examine the Barthel connection and $(\alpha, \beta)$ geometries, where $\alpha$ is a Riemannian metric and $\beta$ is a one-form. For a specific construction of $\beta$, the Barthel connection coincides with the Levi-Civita connection of the associated Riemann metric. We review gravitational field and cosmological evolution in three geometries: Barthel-Randers ($F = \alpha + \beta$), Barthel-Kropina ($F = \alpha^2 \beta$), and the conformally transformed Barthel-Kropina geometry. After presenting the mathematical foundations of Finslerian-type modified gravity theories, we derive generalized Friedmann equations in these geometries assuming a Friedmann-Lema\^itre-Robertson-Walker type metric. We also present the matter-energy balance equations, interpreting them from the perspective of thermodynamics with particle creation. The cosmological properties of Barthel-Randers and Barthel-Kropina models are explored in detail. The additional geometric terms in these models can be interpreted as an effective dark energy component, generating an effective cosmological constant. Several cosmological solutions are compared with observational data (Cosmic Chronometers, Type Ia Supernovae, Baryon Acoustic Oscillations) using MCMC analysis. A comparison with the $\Lambda$CDM model shows that Finslerian models fit observational data well, suggesting they offer a viable alternative to general relativity.